On Zero-Divisor Graph of the ring $\mathbb{F}_p+u\mathbb{F}_p+u^2 \mathbb{F}_p$

Document Type : Original paper

Author

Government Polytechnic College, Sankarapuram, Kallakurichi-606401, Tamil Nadu

Abstract

In this article, we discussed the zero-divisor graph of a commutative ring with identity $\mathbb{F}_p+u\mathbb{F}_p+u^2 \mathbb{F}_p$ where $u^3=0$ and $p$ is an odd prime. We find the clique number, chromatic number, vertex connectivity, edge connectivity, diameter and girth of a zero-divisor graph associated with the ring. We find some of topological indices and the main parameters of the code derived from the incidence matrix of the zero-divisor graph $\Gamma(R).$ Also, we find the eigenvalues, energy and spectral radius  of both adjacency and Laplacian matrices of $\Gamma(R).$

Keywords

Main Subjects

References

[1] D. F. Anderson, A. Frazier, A. Lauve, and P. S. Livingston, The zero-divisor graph of a commutative ring, II, Ideal theoretic methods in commutative algebra, CRC Press, 2019, pp. 61–72.
http://dx.doi.org/10.1201/9780429187902–5
[2] N. Annamalai and C. Durairajan, Linear codes from incidence matrices of unit graphs, J. Inf. Optim. Sci. 42 (2021), no. 8, 1943–1950.
https://doi.org/10.1080/02522667.2021.1972617
[3] N. Annamalai and C. Durairajan, Codes from the incidence matrices of a zero-divisor graphs, J. Discrete Math. Sci. Cryptogr. (In press), https://doi.org/10.1080/09720529.2021.1939955
[4] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Springer, New York, 2012.
[5] R.B. Bapat, Graphs and Matrices, Springer, London, 2014.
[6] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226.
https://doi.org/10.1016/0021–8693(88)90202–5
[7] P. Dankelmann, J.D. Key, and B.G. Rodrigues, Codes from incidence matrices of graphs, Designs, Codes and Cryptography 68 (2013), no. 1, 373–393.
https://doi.org/10.1007/s10623–011–9594–x
[8] I. Gutman and N. Trinajsti´c, Graph theory and molecular orbitals. total Φ-electron energy of alternant hydrocarbons, Chem. Phy. Lett. 17 (1972), no. 4, 535–538.
https://doi.org/10.1016/0009–2614(72)85099–1
[9] T. Kavaskar, Beck’s coloring of finite product of commutative ring with unity, Graphs Combin. 38 (2022), no. 2, 1–9.
https://doi.org/10.1007/s00373–021–02401–x
[10] S. Ling and C. Xing, Coding Theory: A First Course, Cambridge University Press, Cambridge, 2004.
[11] A. Mukhtar, R. Murtaza, S.U. Rehman, S. Usman, and A.Q. Baig, Computing the size of zero divisor graphs, J. Inf. Optim. Sci. 41 (2020), no. 4, 855–864.
https://doi.org/10.1080/02522667.2020.1745378
[12] M. Randić, On chracterization of molecular branching, J. Am. Chem. Soc. 97, no. 23, 6609–6615.
https://doi.org/10.1021/ja00856a001
[13] B.S. Reddy, R.S. Jain, and N. Laxmikanth, Vertex and edge connectivity of the zero divisor graph $\gamma[Z_n]$, Comm. Math. Appl. 11 (2020), no. 2, 253–258.
https://doi.org/10.26713/cma.v11i2.1319
[14] S.P. Redmond, The zero-divisor graph of a non-commutative ring, Internat. J. Commut. Rings 1 (2002), no. 4, 203–211.
[15] R. Saranya and C. Durairajan, Codes from incidence matrices of some regular graphs, Discrete Math. Algorithms Appl. 13 (2021), no. 4, Article ID: 2150035.
https://doi.org/10.1142/S179383092150035X
[16] R. Saranya and C. Durairajan, Codes from incidence matrices of $(n, 1)$-arrangement graphs and $(n, 2)$-arrangement graphs, J. Discrete Math. Sci. Cryptogr. 25 (2022), no. 2, 373–393.
https://doi.org/10.1080/09720529.2019.1681674
 
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 15 October 2023

Files

Share

How to cite

Statistics